Two dueling wizards are at an altitude of $100$ above the sea. They cast spells in turn, and each spell is of the form "decrease the altitude by $a$ for me and by $b$ for my rival" where $a$ and $b$ are real numbers such that $0 < a < b$. Different spells have different values for $a$ and $b$. The set of spells is the same for both wizards, the spells may be cast in any order, and the same spell may be cast many times. A wizard wins if after some spell, he is still above water but his rival is not. Does there exist a set of spells such that the second wizard has a guaranteed win, if the number of spells is $(a)$ finite; $(b)$ infinite?
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mavropnevma
03.05.2011 10:48
Answer to (b) is very easy - YES. Take as set of spells the set $\{S_n = (1/n, 100 - 1/n) \mid n=1,2,\ldots\}$. Once the first wizard uses as his first spell $S_n$, the second replies with $S_{n+1}$. His altitude comes to $(100 - (100 - 1/n)) - 1/(n+1) = 1/n(n+1)$, while the altitude of his rival comes to $(100 - 1/n) - (100 - 1/(n+1)) = -1/n(n+1)$. For (a), let the first player always select a (the) spell with largest difference $b-a$. Thus the altitude of the first wizard will always be at least as large as that of the second, so at no point will it be negative (below water), while the other remains positive (above water). Notice this means not the first wizard wins - the duel may end up with both losing (e.g. when there is only one spell $(40, 45)$).
sabbasi
24.07.2013 19:55
Hello I was recently doing this problem and the official solution is the same as mavropnevma's. I felt like my solution is way too easy (there is no reason to construct any sort of set) but I couldn't find a flaw (but I suspect there is one) so I'm posting it here. The set of spells (a,b) consists of all spells such that 0<a<100 and 0<b<100 with a,b real. Now if the first wizard uses (x,y), then the second wizard responds with (z,100-x) where 0<z<100-y. Such a z is guaranteed to exist and can be made arbitrarily small since we are dealing with real numbers. So the second wizard's height is positive and the first wizards is 0. Does this work? Edit: Wait I think I realized the issue. "Different spells have different values for a and b" means a value of a or b appears once in the set. Sorry